Generating trees for permutations avoiding generalized patterns

TL;DR

This paper constructs generating trees for classes of permutations avoiding generalized patterns, enabling the derivation of functional equations and enumeration results, including refinements and new findings.

Contribution

It introduces a novel tree-based method to enumerate permutations avoiding generalized patterns of length 3 and 4, solving associated functional equations.

Findings

Derived functional equations for permutation classes

Solved equations using kernel method and Bousquet-Mélou's ideas

Refined and discovered new enumerative results

Abstract

We construct generating trees with one, two, and three labels for some classes of permutations avoiding generalized patterns of length 3 and 4. These trees are built by adding at each level an entry to the right end of the permutation, which allows us to incorporate the adjacency condition about some entries in an occurrence of a generalized pattern. We use these trees to find functional equations for the generating functions enumerating these classes of permutations with respect to different parameters. In several cases we solve them using the kernel method and some ideas of Bousquet-M\'elou. We obtain refinements of known enumerative results and find new ones.